Some properties of some functions
Floor function \ (\ lfloor x \ rfloor \ )
（一）\(\lfloor x \rfloor <= x < \lfloor x \rfloor +1\)
(Ii) for any positive integer x and A, B \ (\ lfloor \ lfloor \ FRAC {x} {A} \ rfloor / B \ rfloor = \ lfloor \ {x} {ab & FRAC} \ rfloor \)
(Iii) For any positive integer n, 1 - d n is a multiple of the number of \ (\ lfloor \ frac {n } {d} \ rfloor \)
(Iv) if n is a positive integer, \ (\ lfloor \ FRAC {n} {D} \ rfloor \) a different number of values does not exceed \ (2 \ times \ sqrt { n} species \)
Proof:
\ ((1) If the d \ leq {\ sqrt {n }}, \ lfloor \ frac {n} {d} \ rfloor no more than \ sqrt {n} species \)
\ ((2) if d> \ sqrt {n}, \ lfloor \ frac {n} {d} \ rfloor \ leq \ frac {n} {d} \ leq \ sqrt {n}, \ lfloor \ frac {n } {d} \ rfloor not exceed \ sqrt {n} species \)
\ (In sum, \ lfloor \ frac {n} {d} \ rfloor not more than 2 \ times {\ sqrt {n}} species \)
Harmonic number
Defined \ [Hn = \ sum \ limits_ {k = 1} ^ {n} \ frac {1} {k} \] calculation to obtain \ [Hn = ln (n) + r + o (1) \] wherein r is欧拉马歇罗尼constant, R & lt approximately 0.577, so \ [\ sum \ limits_ {d = 1} ^ {n} \ lfloor \ frac {n} {d} \ rfloor = o (n \ times {logn) } \]
Prime-counting function
\ [\ PI (n) \ SIM \ FRAC {n} {LN (n)} \] \ [so n near the prime number density of approximately \ FRAC {. 1} {LN (n)} \] \ [n-primes pn \ sim n \ times {ln (n)} \]
Number theory function
Multiplicative function
\ [F s function number of positive integers coprime a, b, f (a \ times {b}) = f (a) \ times {f (b}) \]
Completely multiplicative function
\ [F number theoretic function of any positive integers A, B, f (A \ Times {B}) = f (A) \ Times {f (B}) \]
\ (if f is a multiplicative function, \ ) \ [^ {n-A1 = P1} \ {P_2 Times A_2} ^ {} \ {P_3 Times}} ^ {A_3 ...... \ P_s Times {}} ^ {A_S \] \ [F (n- ) = f (p_1 ^ {a_1 }) \ times {f (p_2 ^ {a_2})} \ times {f (p_3 ^ {a_3})} ...... \ times {f (p_s ^ {a_s} )} \]
Unit functions
\[\epsilon(n)=[n==1]= \left\{ \begin{aligned} 1&,n=1\\ 0&,n!=1\\ \end{aligned} \right. \]
Divisor function
\ (\ delta_k (n) represents a k-th power factor and n \)
\ [\ delta_k (n) = \ SUM \ limits_ {D | D ^ k n} \]
Euler function: \ (\ Phi (n-) \)
\ (The Euler function and means no more than n and n is a positive integer prime number \)
Nature:
\ [n = \ SUM \ limits_ {D | n} \ Phi (D) \]
Proof:
\(若gcd(n,i)=d，gcd(\frac{n}{d},\frac{i}{d})=1\)
\ (Where \ frac {i} {d} is not more than \ frac {n} {d} integer, defined by the Euler function, i is the number of \ phi (\ frac {n} {d}) th \)
\ (For all d | n, n = \ sum \ limits_ {d | n} \ phi (\ frac {n} {d}) = \ sum \ limits_ {d | n} \ phi (d) \)
Dirichlet convolution
\ (f, g number on function, the number on the function h satisfies \) \ [h (n-) = \ SUM \ limits_ {D | n-} F (D) G (\ FRAC {n-} {D}) \]
\ ( h is the Dirichlet convolution of f and g is referred to as \)
\ [h = f \ g AST \]
nature
\ ((1) function unit \ epsilon is the Dirichlet convolution unit cell, i.e., for an arbitrary function F, there \)
\ [\ epsilon \ AST = F F \ AST \ epsilon = F \]
\ ((2) commutative and associative \)
\ ((3) If f, g is a multiplicative function, f \ ast g is a multiplicative function also \)
\ ((4) an inverse function: f \ ast f_ inverse = \ epsilon \)
\ (Defined exponential function: Id_k (n) = n ^ k, Id = Id_1 \)
\ (So divisor function: \ delta_k = 1 \ ast Id_k \)
\ (The Euler function: \ phi (n) = 1 \ ast Id \)
Dirichlet convolution calculation:
\ (F, g number theoretic function, then f \ ast g at all about the number n at the need to enumerate the value of n \)
\ (Calculated f \ ast g of the first n items, enumerated 1 to n multiples of each number \)
Mobius function
\[\mu(n)= \left\{ \begin{aligned} &1&n=1 \\ &(-1)^s&n=p_1\times{p_2}\times{p_3}......\times{p_s}\\ &0&otherwise \\ \end{aligned} \right. \]
\(其中p_1,p_2,p_3为素数\)
nature
\[\sum\limits_{d|n}\mu(n)= \epsilon(n) \Rightarrow \mu \ast 1 = \epsilon\]
prove:
\ (N = 1, is clearly established \)
\(n>1\)
\ (N is provided with a different prime factors s, defined by the function Mobius, \)
\ (D if and only if the square-free, \ mu (d)! = 0 \)
\ (So each prime factor index d is 0 or 1 is only \)
\ (So that the binomial theorem \)
\[\sum\limits_{d|n}\mu(d)=\sum\limits_{k=0}^s (-1)^k(s,k)=(1-1)^s=0\]
\ (Is proved \)
Mobius transformation
\ (F number is provided on the function, the function g is defined to satisfy: \)
\ [g (n-) = \ SUM \ limits_ {D |} n-f (D) \]
\ (g is called Mobius transform of f, f is Mobius inverse transform g \)
\ (The Dirichlet convolution \) \ [G = F \ AST. 1 \]
Mobius inversion
\ (G (n) = \ sum \ limits_ {d | n} f (d) is necessary and sufficient conditions for f (n) = \ sum \ limits_ {d | n} g (d) \ mu (\ frac {n } {d}) \)
prove:
\ [g = f \ out 1 \]
\ [f = f \ out \ Epsilon = f \ out 1 \ out \ g = about \ out \ about \]
Proved.
application
Dirichlet convolution can use to solve a series summation problem.
Alternatively a common Dirichlet convolution summation in part, the order of summation and exchange, and ultimately reduce the time complexity.
Common:
\ (\ MU \ AST. 1 = \ Epsilon \)
\ (. 1 Id = \ AST \ Phi \)